In the Quran, two terms are used to refer to those on the straight path (suratal mustakeem): muslim and mumin.

What are some similarities and differences between "muslim" and "mumin"?

In the Quran, it is said that Mumins are the ones who will be salvaged. But I don't remember such a promise for Muslims. Also, I know that the Mumin is coming from the Arabic root "١من" which means "safety". But safety from exactly what?

Are Mumins a subset of Muslims, or are they superset? It is no doubt that these two sets have intersection, but does one of them completely encapsulate the other, or do they have different elements (people)?

6 Answers
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There are two narrations of Hadith Jibreel, when angel Jibreel comes to the Prophet (peace be upon him) in human form and asks him some questions.

These questions include:

What is Islam (who are Muslims?)

What is Eman (who are Mu'mins?)

What is Ihsaan?

The two narrations switch up the order of Islam vs. eman. Therefore, some scholars have said that Islam (basic Islam) is the lower level, and eman (becoming a mu'min) is a higher level of faith. Others say the opposite.

The first opinion is supported by the language. Allah mentions "the ones who believe" in a verb form in many ayaat of Qur'an. In contrast, "mumin" is a noun, a person who believes; someone for whom belief is a strong aspect of their personality.

The [desert] Arabs said: "We believe."
Say: "You haven't believed [yet], but you [should] say: 'We have submitted [to Islamic rule]', for the belief hasn't entered your hearts [yet]. And if you obey God and his messenger, he will not belittle anything from your deeds, for God is oft-forgiving [and] most merciful."

When his Lord said to him [Ibrahim]: "Submit [your will to me]". He said: "I submit [my will] to [you] the Lord of the universe."

It is the same word but the context of these verses shows that it has different meanings.

In the first one it tells those who were claiming to be "believers" that they are not yet, that they have only "submitted" to Islamic rule politically. This is what people mean by "Muslim" means these days, in Quran's terminology they are "followers of the scared laws given to our prophet Mohammad (PBUH)" as "الذین آمنوا" and they are not called "Muslim", "Muslim" in Quran's terminology is a person who has fully submitted to the will of God like Ibrahim, as in the second verse. It is clear that Ibrahim (PBUH) is a believer, but God asks him to submit fully to God's will, which he does.

So when Quran calls a particular person "مسلم" it means that person has submitted to God's will completely like Ibrahim. A "مومن" is a lower level compared to this where one's heart believes in God and hereafter and sacred books and prophets and ..., it doesn't mean the person has reached the the level of "مسلم" as Ibrahim (PBUH) did yet. In this sense the word is not related to following the rules of Islam as a religion but to the higher level concept of submission to the will of God. The word is used for followers of other religions, e.g. in verse 3:52 it is used about Jesus's (PBUH) apostles and in verse 2:136 the sons of Issac (PBUH) say that they are Muslims.

In other contexts, particularly to followers of our prophet Muhammad (PBUH) it sometimes has the first meaning and not the second one (particularly in its verb form "to submit" "أسلم" and not noun "submitted") i.e. the person is submitted to following orders of the prophet and Islamic rules, the person might not be even a real believer at all let alone being a "Muslim" in the sense Ibrahim was.

One should look at the context to see if the intended meaning is the first one (followers of the religion brought by the prophet) or it is the second one (internal characteristic of a person in relation to God's will), and one should also be mindful that how we use these words today might not reflect how Quran uses them.

I quote from The Teachings of Hajj by Shaykh 'Abdur-Razzaaq ibn ‘Abdil-Muhsin al-Badr:

Al-Imaam Ahmad reported in his Musnad that Fadaalah ibn ‘Ubayd narrated that the Messenger of Allaah (Salallahu alayhi wassalam) said during the Farewell Hajj,

“Shall I not inform you about the Mu’min? He is the one whom people
trust with their wealth and lives. The Muslim is he from whose tongue
and hand the people are safe. The Mujaahid is he who struggles with
his own soul to obey Allaah. The Muhaajir is he who abandons misdeeds
and sins.”Musnad Ahmad (6/21); graded saheeh by al-Albaanee in as-Saheehah (549).

This Hadeeth, which is part of the Prophet’s (Salallahu alayhi wassalam) bequest and teachings to his Ummah imparted during the Farewell Hajj, explains the perfection of these
Concepts: Eemaan, Islaam, Jihaad, and Hijrah. It also explains who rightfully deserves to be described by these traits, upon which rests happiness in this world and the Hereafter. Additionally, it gives comprehensive definitions of them.

The Mu’min is he whom people trust with their lives and property. Once Eemaan settles firmly and fills the heart, it demands of a person to fulfill the duties of Eemaan, among the most important of which are: taking care of trusts, dealing with others honestly, and making sure to not oppress others with respect to their lives and property. When someone consistently does these things, people know him to be that way, and they place confidence in him and trust him with their lives and wealth because they know that he
Protects what he is entrusted with.

This is one of the foremost things dictated by Eemaan as the Prophet (salallahu alayhi wassalam) said:

The Muslim is he from whose tongue and hand the Muslims are safe. That is because true Islam is submission to Allah, complete servitude devoted to Him, and giving the Muslims their rights. One’s Islam cannot be complete until he loves for the Muslims what he loves for himself. This can only come about when they are safe from any harm emanating from his tongue and hand, and this is the very root of this obligation. If others are subject to
harm from his tongue and hand, how can he be fulfilling his obligation to his Muslim brothers? How could someone who extends his hand and tongue with harm and enmity against the Muslims possibly be truly practicing Islaam? Thus, their being safe from his verbal or physical harm forms the very essence of completing his Islaam.

This also implies that the Mu’min is a level higher than the Muslim because if someone is trusted with the lives and wealth of others, then the Muslims would obviously be protected from his tongue and hand. Had they not been safe from him to begin with, they would not have trusted him. However, just because they feel safe from him does not mean that they necessarily trust him. He may not harm them, yet they do not place full confidence in him, fearing that he may have withheld his harm due to some ulterior motive or fear, and
not due to having Eemaan in his heart.

Thus, the Prophet (Salallahu alayhi wassalam) defined the Muslim with an outward quality, namely, others being safe from him, whereas he defined the Mu’min with an inner quality, namely, that others trust him with their lives and wealth, and the latter trait is superior to the former.

The Shaykh also similarly explains Mujaahid and Muhaajir but it is not relevant to question. But if you are interested you can read this book - Teachings of Hajj by Shaykh 'Abdur-Razzaaq ibn ‘Abdil-Muhsin al-Badr.

There are many differences between the two. I will be highlighting the major ones.

A Mumin is a higher degree Muslim who:

Repents after every sin (s)he commits.

Feels sorry for every Islamic obligation (s)he either did
not perform or has missed.

Does believe in the message of Allah without the need of concrete evidence (such as supplementary scientific knowledge).

Puts Allah above everything in his/her daily life.

Is and known for being trustworthy.

Is sure of Allah's promise at heart.

Takes into account what Allah prefers.

and so on...

Mystical Approach: A Muslim is a Maqan of Salam (derived from m-slm), and thus carries and spreads Salam, while a Mumin is named directly with God's name. So a standard Muslim is where Salam is observed, while the heart of a Mumin fits Him.

There is lot more to the subject and you may only distinguish between the two clearly if you start taking account the difference between the words when studying the Quran. Most think they are interchangeable, but obviously they are wrong.

Muslim has a general meaning (submission to Allah) and a specific meaning (commonly used now by people around the globe).

The specific meaning still has a large scope as it starts from apparent submission to Allah with minimal state of confession in uniquness of Allah and that the holiness Muhammad --peace be upon him-- in His prophet, to the maximal state of being completely and thoroughly submitted to Allah like the holy prophet was:

Say: "Truly, my prayer and my service of sacrifice, my life and my
death, are (all) for Allah, the Cherisher of the Worlds [6:162]

But being Mu'min is having faith in heart, it's a good state of being Muslim, both in its general and specific meaning of Muslim, so we both have a believer Christian and a believer Muslim. This attribute further measures the degree someone is Muslim (submitted to Allah, in general or in Islam the last version of the heavenly religion):

The desert Arabs say, "We believe." Say, "Ye have no faith; but ye
(only) say, 'We have submitted our wills to Allah,' For not yet has
Faith entered your hearts. But if ye obey Allah and His Messenger, He
will not belittle aught of your deeds: for Allah is Oft-Forgiving,
Most Merciful." [49:14]

although it rather seems that the Christians and other people of book hardly can be considered as Mu'min, most of them can at most be Muslims and at the beginning of being submitted to God, although doing hard worships as well. That's because what they believe in God and His prophet is close to Kufr and Shirk. And if they remedy such ideas then they will commonly convert to Islaam and if they don't live much after remedying their beliefs to convert and die they might be still considered as Mu'min (even although not converted according to the rules of Islam). I rather prefer to skip this issue as I have very small evidences on this issue, and Allah knows best. But let bring these verses that may help:

O People of the Book! Commit no excesses in your religion: Nor say of
Allah aught but the truth. Christ Jesus the son of Mary was (no more
than) a messenger of Allah, and His Word, which He bestowed on Mary,
and a spirit proceeding from Him: so believe in Allah and His
messengers. Say not "Trinity": desist: it will be better for you:
for Allah is one Allah: Glory be to Him: (far exalted is He) above
having a son. To Him belong all things in the heavens and on earth.
And enough is Allah as a Disposer of affairs. [4:171]

And there are, certainly, among the People of the Book, those who
believe in Allah, in the revelation to you, and in the revelation to them, bowing in humility to Allah: They will not sell the Signs of
Allah for a miserable gain! For them is a reward with their Lord, and
Allah is swift in account. [3:199]

Among the People of the Book are some who, if entrusted with a hoard
of gold, will (readily) pay it back; others, who, if entrusted with a
single silver coin, will not repay it unless thou constantly stoodest
demanding, because, they say, "there is no call on us (to keep faith)
with these ignorant (Pagans)." but they tell a lie against Allah, and
(well) they know it. [3:75] [and honesty, truthfulness and
trusteeship are among the most important measures of Iman according to
Ahadeeth, even more important measures than prayer and fast and etc.]

And at a last point, Iman itself has several ranks and it will be measured by Taqwa if I'm not wrong:

O ye who believe! Believe in Allah and His Messenger, and the scripture which He hath sent to His Messenger and the scripture which
He sent to those before (him). Any who denieth Allah, His angels, His
Books, His Messengers, and the Day of Judgment, hath gone far, far
astray. [4:136]

O ye that believe! Fear Allah, and believe in His Messenger, and He will bestow on you a double portion of His Mercy: He will provide
for you a Light by which ye shall walk (straight in your path), and He
will forgive you (your past): for Allah is Oft-Forgiving, Most
Merciful.

There are many verses that Allah tells the believers to believe, to fear Allah, and warn them, but there is no where that Allah warns a Mutaqqi who fears Allah as this fear is itself a mean to save the Mu'min.

PS. there may exists gaps in this answer, so feel free to fill them. Also bear in mind that the writer of the answer is not a scholar/expert and can easily make mistakes, although tries to prevent errors.

My dear brothers & sisters, don't get confused. It’s very simple,,, I can give You supports and evidences , but right now You all have heard as posted above...

First two and last comments are completely wrong...

ISLAM is to completely submit ones will to Allah

Well now the Beginning Muslim has to believe in Allah and do good, just like everyone who lived before the Quran... It doesn't mean they haven't believed, they have believed in Allah...

But basically there are three levels of Faith...

ISLAM with five pillars of faith
IMAN with Deep Belief in 6 principles
IHSAN which means excellence

So basically anyone who testifies is a Muslim, means has submitted

But to become a true Muslim, he needs to submit his will completely,, yes, he needs to become a Muslim, then a MUMIN and a Muhsin,,,, So actual speaking a true Muslim is one who have acquired three levels of Faith,, ( do basics of ISLAM, supported by true belief or IMAN and finally excel as righteous Muslim or have IHSAN )

IN SHORT A BASIC MUSLIM WHO SIMPLY SUBMITED WILL BECOME A TRUE MUSLIM WHO HAS COMPLETELY SUBMITTED OR SURRENDERED...

OR U CAN SAY A BASIC MUSLIM BECOMES EXCELLENT MUSLIM,,, and there lies the wisdom in the religion of faith...

Never get confused reading Quran oneself... Allah has asked to follow Sadiqeen, Saliheen, Mutaqeen and the Mumineen if You have confusions...

See, I explained in simple by the will of Allah English because I follow and learn from a obedient Saadiq of Allah who is also related to the family of Muhammed Rasool (SAWS) .

In Shaa Allah... jazakallah khair

(Never write insha Allah, because it means WE CREATED ALLAH... so write In-Shaa-Allah)